Kantorovich-Rubinstein Distance and Approximation for Non-local Fokker-Planck Equations

TL;DR

This paper investigates the behavior of solutions to non-local Fokker-Planck equations driven by Levy noise, providing distance estimates, convergence results, and smooth approximations for these complex stochastic systems.

Contribution

It introduces new estimates for the Kantorovich-Rubinstein distance and demonstrates weak convergence and smooth approximation methods for non-local Fokker-Planck equations with Levy noise.

Findings

Established bounds for the Kantorovich-Rubinstein distance

Proved weak convergence of probability distributions

Developed smooth approximation techniques for non-local equations

Abstract

This work is devoted to studying complex dynamical systems under non-Gaussian fluctuations. We first estimate the Kantorovich-Rubinstein distance for solutions of non-local Fokker-Planck equations associated with stochastic differential equations with non-Gaussian Levy noise. This is then applied to establish weak convergence of the corresponding probability distributions. Furthermore, this leads to smooth approximation for non-local Fokker-Planck equations, as illustrated in an example.